Journal of Determining the evolution of an alpine drainage system by solving inverse problems

Inigo Irarrazaval1 , Mauro A. Werder2,3 , Matthias Huss2,3,4 , 1 1 Article Frederic Herman and Gregoire Mariethoz

1 Cite this article: Irarrazaval I, Werder MA, Huss Institute of Earth Surface Dynamics, Faculty of Geosciences and Environment, University of Lausanne, Lausanne, M, Herman F, Mariethoz G (2021). Determining ; 2Laboratory of Hydraulics, Hydrology and Glaciology (VAW), ETH Zurich, Zurich, Switzerland; 3Swiss the evolution of an alpine glacier drainage Federal Institute for Forest, Snow and Landscape Research (WSL), Birmensdorf, Switzerland and 4Department of system by solving inverse problems. Journal of Geosciences, University of Fribourg, Fribourg, Switzerland Glaciology 67(263), 421–434. https://doi.org/ 10.1017/jog.2020.116 Abstract Received: 12 June 2020 Our understanding of the subglacial drainage system has improved markedly over the last decades Revised: 16 December 2020 Accepted: 17 December 2020 due to field observations and numerical modelling. However, integrating data into increasingly First published online: 22 January 2021 complex numerical models remain challenging. Here we infer two-dimensional subglacial chan- nel networks and hydraulic parameters for , Switzerland, based on available field Key words: data at five specific times (snapshots) across the melt season of 2005. The field dataset is one of Glacier modelling; glaciological instruments and methods; subglacial processes the most complete available, including borehole water pressure, tracer experiments and meteoro- logical variables. Yet, these observations are still too sparse to fully characterize the drainage sys- Author for correspondence: tem and thus, a unique solution is neither expected nor desirable. We use a geostatistical Inigo Irarrazaval, generator and a steady-state water flow model to produce a set of subglacial channel networks E-mail: [email protected] that are consistent with measured water pressure and tracer-transit times. Field data are used to infer hydraulic and morphological parameters of the channels under the assumption that the location of channels persists during the melt season. Results indicate that it is possible to identify locations where subglacial channels are more likely. In addition, we show that different network structures can equally satisfy the field data, which support the use of a stochastic approach to infer unobserved subglacial features.

1. Introduction Water flow at the ice/bedrock interface exerts controls on ice flow (e.g., Iken, 1981), glacier erosion (e.g., Herman and others, 2011), sediment transport (e.g., Beaud and others, 2018), as well as catchment hydrology (e.g., Verbunt and others, 2003). Hence, a significant scientific effort has been dedicated to characterizing the spatial distribution of subglacial systems and their temporal evolution by both field data acquisition and numerical modelling. Pioneer work conceptualized the subglacial drainage system with two main components: a network of subglacial channels with fast water flow (Röthlisberger, 1972;Nye,1976) and a distributed system with the slow flow (Lliboutry, 1968; Weertman, 1972). In addition, it has been observed that areas of a subglacial drainage network can become temporally or permanently hydraulic- ally disconnected, which can be described as the third component of subglacial systems (e.g., Iken and Truffer, 1997; Hoffman and others, 2016; Rada and Schoof, 2018). The study of subglacial systems has focused on both field observations and numerical mod- elling. Field techniques that have been employed include water level measurements in bore- holes drilled into the glacier (e.g., Hubbard and others, 1995; Rada and Schoof, 2018), dye-tracer tests in moulins (e.g., Nienow and others, 1998; Schuler and others, 2004; Werder and others, 2009; Chandler and others, 2013) and glacial speleology expeditions to survey portions of channel networks (e.g., Gulley and others, 2014). More recently, seismic and other geophysical methods have also provided insights into the evolution and structure of the subglacial systems (e.g., Gimbert and others, 2016; Church and others, 2019; Zhan, 2019; Lindner and others, 2020; Nanni and others, 2020). In addition to field methods, a range of subglacial numerical models have been built, which we divide here into two categories: (1) subglacial routing models and (2) subglacial evolution © The Author(s), 2021. Published by models. Routing models need to make an assumption for the hydraulic potential to compute Cambridge University Press.. This is an Open Access article, distributed under the terms of the water flow direction. Most commonly, the water pressure is assumed equal to the ice over- the Creative Commons Attribution- burden pressure to which the elevation potential is added to get the hydraulic potential, as was NonCommercial-ShareAlike licence (http:// first proposed by Shreve (1972). Then, a routing algorithm (e.g., O’Callaghan and Mark, 1984; creativecommons.org/licenses/by-nc-sa/4.0/), Tarboton, 1997) calculates a map of the upstream area or accumulated flow, where locations which permits non-commercial re-use, with higher upstream areas are conceptualized as channels. Routing models have been widely distribution, and reproduction in any medium, provided the same Creative Commons licence used to study subglacial systems, with applications including channel network topology deter- is included and the original work is properly mination (e.g., Richards and others, 1996), the occurrence of subglacial lakes (e.g., Livingstone cited. The written permission of Cambridge and others, 2013) and delineation of subglacial basins (e.g., Chu and others, 2016). University Press must be obtained for Furthermore, Banwell and others (2013) and Arnold and others (1998), constructed a channel commercial re-use. network structure based on routing methods on which they then calculated the water pressure and channel radii evolution. One of the limitations of such routing models is that only one single subglacial channel network is obtained and then further used for all numerical experi- cambridge.org/jog ments. Moreover, when combining the surface elevation and bedrock elevation, different

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sources and spatial scales must be either downscaled or upscaled water pressure from boreholes is used to condition the subglacial to reach a common resolution, which in turn adds uncertainties. system through an inversion procedure. In addition, tracer-transit Efforts to explore the variability of the subglacial network struc- times from dye-tracer tests carried out from moulins are used to tures have considered the effects of varying a spatially uniform constrain the water velocities in the subglacial channels to a phys- flotation factor in Shreve’s equation. They have found that chan- ically reasonable range. nel topology and subglacial basin delineation are sensitive to the The remainder of this paper is organized as follows. Section 2 choice of a flotation factor (e.g., Chu and others, 2016). A first describes the model framework, which has its roots in Irarrazaval effort to consider a spatially two-dimensional (2-D) nonuniform and others (2019), however, incorporating new adaptations to fit term which adds variability to the channel network was presented real-world applications. Section 3 describes the field site, datasets, in Irarrazaval and others (2019). However, it was tested in a syn- modelling choices and assumptions to apply the framework to the thetic setup and used to infer the channel structure and hydraulic Gorner Glacier. Section 4 shows a set of subglacial systems condi- properties rather than the actual channel location. tioned to data across the melt season and parameter uncertainties The second category of models represents, besides water flow, provided by the inversion. The last section discusses the contribu- the process of the evolution of the subglacial drainage system. tion and limitations of a physical-geostatistical subglacial model First, 1-D models represent the evolution of one subglacial chan- and trade-offs in integrating data in complex numerical models. nel, for example, for describing glacial lake outburst floods (e.g., Flowers and others, 2004). More recently, 2-D subglacial evolu- tion models became capable of the spontaneous formation of 2. Methods channel network structures as the outcome of simulating subgla- The methodology consists of four steps. First, we use a geostatis- cial drainage system equations. They successfully incorporated tical subglacial channel generator to produce a set of uncondi- many of the known physical processes, including channel opening tioned subglacial systems states for a reference snapshot. The (by heat dissipation) and closing (by viscous-ice creep) allowing subglacial channel generator approximates the hydraulic potential the formation and evolution of channels and distributed systems by Shreve’s equation solely to produce channel networks based on (e.g., Schoof, 2010; Hewitt, 2011; Werder and others, 2013; Rada a flow routing approach (see Section 2.1). Second, we compute the and Schoof, 2018). However, in real-world applications, such hydraulic potential and water discharge inside the unconditioned approaches are difficult to be implemented and often cannot pro- subglacial systems, consisting of the above-generated network and duce models that match observations. Exceptions include the a homogeneous distributed system, by a water flow model. The probabilistic inference of a lumped glacier hydrology model by water flow model solves laminar/turbulent flow equations in Brinkerhoff and others (2016), in addition to Koziol and steady-state conditions to recover water pressure and discharge Arnold (2017) with uses an optimization approach to infer certain in the channel network (see Section 2.2) and the distributed sys- model parameters from data. tem. Third, a formal parameter inversion is performed, which However, the inversion is applied to a lumped (zero- enables assessing the misfit of the modelled water pressure and dimension) glacial hydrology model, and, hence, the channel net- tracer-transit times with field observations by a likelihood func- work is not explicitly modelled. In any case, it is not possible to tion. The outcome is a set of subglacial systems conditioned to validate the produced channel networks due to the restricted data for the reference snapshot (see Section 2.3). Lastly, for the access to the subglacial drainage systems because these models subsequent snapshots across the melt season, the temporal per- often require a high number of parameters whose uncertainty sistence of the channel network throughout the season is guaran- cannot be fully constrained by observations (de Fleurian and teed by allowing only small perturbations to the maximum others, 2018). likelihood subglacial system of the reference snapshot. The out- The aim of the present paper is to infer, based on field obser- come is a set of subglacial systems honouring the observations vations and prior knowledge, the location of subglacial channel during each snapshot, which allow assessing the changes in the networks and their hydraulic parameters at different snapshots system across the melt season (see Section 2.4). The core of the during the melt season. As information is scarce and contains methodology is based on Irarrazaval and others (2019), with uncertainties, a unique solution is neither possible nor desirable, two significant improvements: (1) the subglacial channel gener- and the model has to be formulated as an inverse problem using a ator has been expanded to account for channel location. (2) probabilistic approach based on Bayesian inference. The proposed The parameter inference strategy has been adapted and improved framework is based on a modified version of the method intro- to suit the application to a real-world dataset and to assess the duced by Irarrazaval and others (2019). It allows inferring the development of the channel network across the melt season. main characteristics and hydraulic parameters of subglacial chan- nels from real data, in turn enabling the assessment of the channel-network development across the melt season. The mod- 2.1 Subglacial channel generator elling approach includes a parsimonious, statistics- and a physics- based model which is computationally inexpensive, allowing us to The subglacial channel generator is a statistics- and physics-based perform a large number of model runs and, thus, enabling uncer- tool that creates a channel network embedded in a 2-D distributed tainty quantification via Markov chain Monte Carlo methods. system (Irarrazaval and others, 2019). It depends on seven para- The framework explores a range of different network structures meters whose values will be inferred in our inversion procedure. by including a spatially nonuniform perturbation of the hydraulic The subglacial channel generator presented in this study incorpo- potential used in the subglacial routing method. Comparison with rates two novelties: (1) channel location variability is incorporated field data are achieved by using a water flow model that provides by extending the parametrization and adding a geostatistical par- water pressure and tracer-transit time. ameter. (2) The flotation factor f in Shreve’s equation (Shreve, The framework is applied to the trunk of the Gorner Glacier, 1972) is incorporated to add more variability to the channel Switzerland, during the 2005 melt season. We consider five snap- networks. shots during which the hydraulic setting is considered constant in Note that the subglacial channel generator produces a set of terms of hydraulic conditions, meltwater input and topology of channel networks embedded in a distributed system. Water pressure the subglacial system. Our goal is to determine the subglacial and flow speed in the subglacial systems is then computed by a sep- drainage system for each of these snapshots. For each of them, arate model via flow equations and recharge forcing (Section 2.2).

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The subglacial systems are set up in a 2-D domain considering threshold parameter c correspond to the channel network. pressurized water flow, where water follows the hydraulic poten- Parameter c is set to ensure that all moulins are connected to tial defined by the subglacial channels. Flow routing and channel structure are computed using TopoToolbox 2 (Schwanghart and Kuhn, 2010; = + . ( ) f fz pw 1 Schwanghart and Scherler, 2014). The last step is to assign radii to each of the channel network ϕ ρ ’ Here pw is the water pressure and the z = wgB is the elevation segments ri. This is done computing the Shreve s stream order ui ρ potential with water density w, acceleration of gravity g and bed- (Shreve, 1966) and transforming it to a radius using the following rock elevation B(x,y). equation (Borghi and others, 2016), The first step is to generate the channel network topology or uib channel architecture. Similar to previous studies based on routing r(ui) = ae , (3) algorithms (e.g., Arnold and others, 1998; Chu and others, 2016), Shreve’s approximation to the hydraulic potential (Shreve, 1972) where a and b are parameters determined in the inversion proced- ρ is obtained by setting pw = fpi in Eqn (1). Where pi = igH is the ure. Then, the channel network is embedded into the distributed ρ ice overburden pressure considering an ice density i and ice system modelled as a homogeneous 2-D layer with transmissivity thickness H(x,y). The flotation factor f is usually set to 1 but Td. Lastly, in order to model dye-tracer transit speeds, we account values ranging from 0.6 to 1.1 have been employed previously for the moulins’ inflow modulation time. Inflow modulation time (e.g., Chu and others, 2016). Here, we include a nonuniform corresponds to the time water spends in the englacial drainage ϕ ’ noise or perturbation field R(x, y) which is added to Shreve s system until it reaches the subglacial channels. Moreover, inflow hydraulic potential to obtain an approximated hydraulic potential modulation time, which has been shown to significantly affect field ϕ*, the overall tracer transit times (e.g., Werder and others, 2010), could be estimated if the recharge hydrograph, subglacial water ∗= + + . ( ) f rwgB f rigH fR 2 pressure and moulin geometry are available. As these data are τ unavailable for the study site, we include parameter i which ϕ The role of R is key in determining channel location. accounts for the inflow modulation time where the subscript i Flow routing can be highly sensitive to small variations in the enumerates the moulins with tracer data. potential surface and a small perturbation could induce large In summary, the channel generator uses seven parameters. ϕ downstream deviations of hydraulic paths. In this study, R is Parameters a and b determine the channel radii and Td the trans- modelled as a Gaussian random field (GRF). The GRF is a spa- missivity of the distributed system. The structural parameters lxy, s tially correlated random field generated from random white and the flotation factor f control the channel location and/or net- noise and parametrized in terms of its variance and integral work topology. The first two modify the GRF, whereas the flota- scales. This is useful, as varying the variance, integral scales tion factor controls an overall trend on Shreve’s equation. Finally, τ and modifying the white noise allows for perturbations of dif- i accounts for the inflow modulation time. All parameters are ferent magnitude, correlation in space and location, which inferred in the inversion procedure. influence the channel topology. However, one of the challenges is to gradually modify the white noise to provide a continuous transition between channel networks. To overcome this issue, ϕ 2.2 Water flow model we create a larger GRF from which we crop R by varying the shift s of the cropping-region along the y dimension (Fig. 1). A water flow model is used to compute water pressure pw (Eqn Tests showed that the overall channel location probability is (1)) and water discharge inside previously generated subglacial not affected by the dimension along which s is varied. This is systems. Water flow is solved in steady-state conditions using expected as the GRF is isotropicandlargeenoughtoaccommo- the finite element GROUNDWATER code (Cornaton, 2007) date several independent cropping-regions or realization of ϕR. based on the framework presented in Irarrazaval and others The inclusion of s allows for a smoother deformation of the (2019). The subglacial drainage system is modelled as two network, useful when exploring the model space (see in coupled components: distributed and channelized. Both compo- Supplementary material: Subglacial channel generator video nents of the subglacial system are coupled using a finite-element 1). As an analogy, the variance, integral scales and shift can mesh with shared nodes, assuming continuity of the pressure be seen as changing the amplitude, wavelength and phase field. This allows water and mass exchanges between the distrib- shift of a periodic wave, which results in a simple paramet- uted system and channels that are driven by the pressure rization of the GRF. Note that Irarrazaval and others (2019) gradient. found that a sufficient channel network variability can be The distributed system is modelled as a 2-D layer, discretized achieved with a fixed GRF variance. Here we chose a GRF vari- in 25 m edge quadrangles and with homogeneous transmissivity 2 ance of 0.0058 (MPa) equivalent to a standard deviation of Td. Water mass is conserved assuming incompressibility and pres- ∼7.7 m water column. The selected value is similar to the surized flow, then Darcy-laminar flow is considered under the small-scale variations of ϕ,suchthatitinfluencesϕ* but does assumptions of a nondeformable porous medium and not disrupt the large-scale trend. Then, the subglacial channel vertically-integrated flow equations: generator considers three parameters that modify ϕ*: flotation =− ∇ ( ) factor f, integral scales lxy and shift s, hereafter referred to as q Td f, 4 the structural parameters of the subglacial channel generator. Once the hydraulic potential is obtained, the D8 flow routing with q the flux, Td the transmissivity of the distributed system and algorithm (O’Callaghan and Mark, 1984) is computed over the ∇f the gradient of the hydraulic potential. hydraulic potential ϕ* and the upstream contribution flow accu- The channelized system is modelled as a network of 1-D cylin- mulation is computed for each cell. In this step, the spatial varia- drical elements of radius r. Note that the radii are fixed as we are tions in water recharge (i.e., moulins and water recharge by considering a snapshot model (no time derivatives). Discharge in tributary ) are considered to compute the upstream flow the channel network is computed using the nonlinear contribution. The flow paths with an accumulation higher than Manning-Strickler empirical formulation for turbulent flow

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Fig. 1. Synthetic example of subglacial channel gener- ator (water flow from right- to left-hand side). On the right-hand side colour fields correspond to the GRF

(ϕR), and on the left-hand side to unconditioned subgla- cial channels. The influence of the shift s parameter is

shown in (a), (b) and (c), by shifting the GRF ( f and lxy are fixed) and computing the channels. (a) and (b) show minor differences when the shift is small. (c) shows different structures as it considers a completely different part of the GRF. In (d), shift is the same as in

(c), but the integral scale (lxy) of the GRF is smaller.

… assuming mass conservation, incompressibility and pressurized run using m as parameters. Field data d=[d1, dn] correspond to flow. The water flow inside channels Q is computed by the observations, with n the total number of data. The standard σ σ …σ deviation of the errors =[ 1, n] have to be estimated and Q =−K∇f, (5) should reflect not only instrumental errors, which are relatively small, but also the effect of averaging a point measurement over with a model cell and model errors. The choice of the uncertainties is a subject of debate; for example, Brinkerhoff and others / 2/3 (2016) chose uncertainties similar to the magnitude of daily var- = a(r√i 2)  ( ) K , 6 iations. However, the setup here differs as the inversion is per- nm |∇f| formed for a snapshot of time. We run several inversions to test σ where K corresponds to the channel hydraulic conductivity, with the effect of in the likelihood function and select a value that α π 2 −1/3 avoids overfitting but represents the uncertainties that each data a circular cross-section = ri , and nm =0.04m s is the Manning friction coefficient for subglacial channels (e.g., Gulley source is subjected to, such as field measurement errors, model and others, 2014). errors and limitations (e.g., mesh resolution). The model is set up with prescribed water recharge from the Each data source is subjected to field measurement errors, tributaries and moulins that enters via subglacial channels. In which are added to model errors and limitations (e.g., mesh reso- addition, discharge at the glacier’s outlet is modelled as atmos- lution). Note that Eqn (9) is a general formulation that is subse- pheric pressure Dirichlet boundary condition. The total residence quently adapted for each kind of data, as is commonly done for times of tracers are calculated in two steps. First, tracer-transit joint inversion. Assuming independence, the total likelihood is times are computed by integrating the advective velocity along the product of the individual likelihoods, which can be stated as the subglacial channels and secondly, by adding the inflow modu-  τ lation time ( i), which is a parameter to be inferred.  L(m|d) / exp − Sj , j [ {pw, ts, z}. (9) j 2.3 Parameter inversion strategy The inversion aims to find the subglacial networks and hydraulic Here each misfit function Sj is designed to account for the par- parameters that enable the model to fit the observations. As data ticularities of the observational data type, where Spw, Sts and Sz are scarce and uncertain, we expect the solution to be highly non- correspond to water pressure observations, tracer-transit speeds unique, meaning that many networks can honour the observa- and a correction term to account for piezometric level exceeding tions. Therefore, we use a maximum likelihood approach where the surface elevation (see below). the goal is to explore the model space and find multiple condi- Borehole water pressure data are subject to different sources of tioned subglacial (CS) systems. Here, we define a likelihood func- uncertainty. Furthermore, it has been observed that neighbouring tion, which is a measure of the misfit of the field data with the boreholes can show significant differences in water level and model outputs, to evaluate model parameters. A general formula- behaviour (e.g., Rada and Schoof, 2018). Therefore, we have low tion of a likelihood function, assuming independent Gaussian confidence in the spatial representativeness of a measurement. uncertainties for the data (e.g., Mosegaard and Tarantola, 1995) The latter is implemented in the likelihood function by adding is given by a tolerance to the borehole location, i.e., we select the modelled value within a radius rbh which is closest to the measurement L(m|d) / exp(−S), (7) and use this in the likelihood. Then, the misfit function is stated as: where  2 n | − pw |  1 min( Fi(m) d ) n − 2 S = i,g , 1 Fi(m) di pw pw S(m) = , (8) 2 i=1 si 2 si i=1 [ ∀ − 2 + − 2 , 2 ( ) g { xg, yg with (xi xg) (yi yg) rbh }, 10 τ with S the misfit function, m =[a, b, lxy,s,f,Td, i] the model parameters and F the forward model (water flow model and sub- where subscript γ refers to the neighbouring cells (coordinates glacial channel generator). The forward model outputs F(m) xγ, yγ) within a distance rbh of a borehole location (xi,yi). The pw (water pressure and tracer transit times) are based on a simulation standard deviation of borehole data are considered as si .

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Tracer-transit times provide an estimate of subglacial water 2.4 Channel network structure persistence across the melt flow conditions. However, the total tracer resident times are season strongly influenced by the inflow modulation time (time spent The proposed methodology finds sets of subglacial systems (or inside the moulin), which is estimated from water recharge into model parameters) for one snapshot. We refer to the set of the moulin and its geometry. Within the data used in this inferred subglacial systems as CS systems, as they honour the study, there are no observations about recharge conditions into observations. While this is acceptable for a single snapshot, if moulins. Consequently, observed transit times are only used to independent inversions are carried out for different snapshots, constrain or regularize the model into a range of physically plaus- the spatial and topological dependence of the channel network ible transit speeds (Schuler and others, 2004; Werder and others, structure across the melt season is not guaranteed. Similar to pre- 2010). We define the binary variable I ts as an indicator of the vious studies (Arnold and others, 1998; Banwell and others, tracer-transit speeds within a physically reasonable range defined 2013), we assume that the channel network structure does not by dts and dts . Transit speeds outside the physically reasonable min max change radically in the course of the melt season. While, changes range are then penalized through a likelihood function with in recharge during the course of the season can affect the standard deviation sts: i hydraulic parameters and channel radii. To ensure channel-architecture dependence between snap- n | − ts | | − ts | 2 = 1 Its min( Fi(m) dmin , Fi(m) dmax ) shots across the melt season, we force the parameters that define Sts ts , 2 i=1 si structural parameters of the subglacial channel generator (lxy, s  (11) and f ) to remain within a vicinity. For this, we implement the 0, dts ≤ F (m) ≤ dts , Its = min i max inversion in two steps: First, we infer a set of subglacial systems 1, dts . F (m)ordts , F (m). conditioned to a reference snapshot (labelled with subscript R: min i max i τ aR,bR,lxy,R,sR,fR,Td,R, i,R), select manually one model (e.g., maximum likelihood mode) and extract the structural parameters Additionally, we assume that it is unlikely that the hydraulic ’ ’ ’ (labelled with prime: lxy ,s,f). The second step consists in infer- head exceeds the surface elevation. To this end, we implement ring independently the subglacial systems of all other snapshots sϵ an equation that is activated only if water pressure exceeds surface {1, …, 5}, limiting the exploration of the structural parameters to elevation and penalizes the likelihood when the water level is ’ ’ ’ the vicinity of the selected model (lxy ,s,f). The bounds are set in higher than the surface elevation. This is implemented as a trun- order to allow perturbations such as channel translation and local cated Gaussian given by deformation to avoid large network reorganizations. This is done ’ ’ ’  by sensitivity tests that constrain the bounds of lxy ,s and f n − z 2 ≤ = 1 Iz Fi(m) i Iz = 0, Fi(m) zi ( ) such that no large reorganizations are observed in the network. Sz z , . 12 2 = s 1, Fi(m) zi, As a result, the inferred structural parameters for each snapshot i 1 i ’ ’ ’ (l,xy,s ,ss ,fs ) are near the selected model. The inferred structural parameters for all snapshots are therefore correlated because they where I z is an indicator corresponding to the index of nodes are forced to be in the surroundings of the same local maximum where F (m) has exceeded the surface elevation z with a standard i i of the likelihood function, imposing a dependence across the melt deviation sz. i season. Note that the channel radius parameters (a and b), trans- One of the challenges in the model inversion is that we expect missivity T of the distributed system and τ are not constrained to different channel networks with the correct hydraulic parameters d i remain in the vicinity of the selected model (Fig. 2). to have a similar hydraulic response (i.e. to honour the observa- tions). In other words, we expect to find several local maxima, which can be distant in the model space. In addition, there are nonlinearities and discontinuities in the likelihood function. For 3. Application to the trunk of the Gorner Glacier example, gradually modifying structural parameters (l ,s,f) xy 3.1. Field site and dataset can modify the channel network until a large reorganization is introduced. This is, for instance, the case when two parallel chan- The study site and model area correspond to the trunk of Gorner nels are being gradually pushed together until they merge, cul- Glacier, Switzerland (Fig. 3). Gorner Glacier is located in the minating in a larger restructuring of the network. at the foot of and has experienced sub- Consequently, we choose an inversion algorithm that explores stantial mass loss since the 1980s (Huss and others, 2012). The the space thoroughly and avoids getting trapped in one local max- glacier system consists of five tributaries covering an area of imum. This is commonly done by Markov chain Monte-Carlo ∼50 km2 (in 2003). The model extent comprises the ablation algorithms such as the Metropolis-Hasting algorithm (Metropolis zone of the main glacier tongue, with an area of ∼6.5 km2 and and others, 1953). Here, we use a more advanced algorithm, the a surface elevation range between 2200 and 2700 m a.s.l.. Recent −1 Markov chain Monte-Carlo DREAM(ZS) algorithm (Laloy and observations indicate surface velocities of up to 200 m a near Vrugt, 2012). DREAM(ZS) explores the model space similarly to a the confluence of the tributaries Grenz and Zwillings Glacier, − Metropolis-Hasting algorithm but includes multiple chains an adap- 35 m a 1 near bh1 (Fig. 3) but much slower surface movement tive sampling which facilitates a robust exploration of the model over much of the lower part of the glacier (Benoit and others, space and therefore it is able to find several local maxima more 2019). In addition, bedrock elevation maps show an extended readily. In addition, we configure DREAM(ZS) with different starting over-deepening where ice thickness exceeds 400 m (Sugiyama points to ensure an extensive model space exploration. As a result, and others, 2008). A distinct feature of the Gorner Glacier is DREAM(ZS) captures the multimodal distribution of our problem, the polythermal composition in the ablation zone. A cold ice where each mode corresponds to a local maximum and therefore body extends between the Grenz and Gorner confluence (near to a CS network. Moreover, it allows uncertainty quantifications boreholes bh1–7 expecting bh5 in Fig. 3a), occupying 80–90% of the model parameters for each explored local maximum. Note of the ice thickness (Ryser and others, 2013). The cold ice is that this involves running the inversion procedure numerous impermeable to meltwater, except for fracturing or water flowing times, which is feasible as our model is relatively computationally permanently through cracks and moulins, leading to supraglacial inexpensive. streams and ponds.

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Fig. 2. Parameter inversion strategy and snapshot dependence. First, the parameters for a reference snapshot are inferred and one model is selected (e.g., maximum likelihood model). For the following snapshots, the structural parameters are con- strained to the vicinity of the structural parameters from the selected model.

Fig. 3. (a) Model configuration, the location of boreholes (bh), and moulins (M) with tracer test data. (b) Overview of the Gorner Glacier system in 2005. The inves- tigated area is shown in grey. Abbreviations correspond to the tributary glaciers Unterer Theodul (The), (Bre), Schwärze (Sch), Zwillings (Zw), Grenz (Gre), and Gorner (Gor) Glacier.

Gorner Glacier was extensively studied over the years 2004–08 Water levels are available from seven boreholes drilled down to with the main focus in glacial lake outburst floods, which occurred the glacier bed and instrumented with pressure transducers periodically from the drainage of the Gornersee. Extensive field- (Riesen and others, 2010). Boreholes bh1, bh3, bh4 show strong work and data collection were done providing one of the most com- diurnal amplitudes of water head (sometimes in excess of 100 m) plete of such datasets (e.g., Huss and others, 2007;Sugiyamaand almost in phase with the discharge measured at the gauging station others, 2008; Walter and others, 2008; Werder and others, 2009; (Fig. 3, Fig. 4). Conversely, boreholes bh5, bh6 and bh7 show minor Riesen and others, 2010). For this study, we use datasets acquired or no diurnal oscillations across the season, with a relatively higher in 2005 which include borehole water pressure data (Huss and water level. Note that at the beginning of August 2005, mean water others, 2007; Riesen and others, 2010), dye-tracer experiments levels rose and oscillations reduced significantly in boreholes bh1 (Werder, 2009; Werder and others, 2009) and discharge of the and bh4. Borehole bh2 shows an anticorrelated behaviour with an catchment. In addition, hourly surface melt on a 25 m grid is avail- amplitude of 3 m. Such an anticorrelated signal has been attributed able from a melt model calibrated to in situ ablation measurements to stress re-distribution at the glacier bed (e.g., Murray and Clarke, and catchment discharge (Huss and others, 2008). Estimates of ice 1995; Lefeuvre and others, 2018; Rada and Schoof, 2018). Overall, thickness distribution are available for the entire glacier system boreholes with a steady and relatively high-water level can be inter- based on ground-based and airborne ground-penetrating radar sur- preted as situated in partly or fully disconnected areas, whereas veys and spatial interpolation of the radar profiles (Farinotti and boreholes with strong diurnal-oscillating water level are situated in others, 2009; Rutishauser and others, 2016). Furthermore, repeated hydraulically well-connected areas. In addition, the rise in the DEMs providing information on ice surface topography are avail- water level of bh1 and bh4 in August 2005 can be interpreted as able (Bauder and others, 2007). Key aspects of the datasets are sum- a transition from a channelized to a distributed or less efficient sys- marized below and illustrated in Figures 3, 4. tem in their vicinity.

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a

b

Fig. 4. (a) Observed borehole water levels, measured outlet discharge and modelled discharge. Tracer-transit times correspond to vertical green bands defined by the injection time and the time of exit at the outlet (for reference, thicker line is ∼13 h and thinner range from 2 to 4 h). Each modelled snapshot is denoted by black dashed line. (b) Zoom for each snapshot.

Discharge for the entire catchment is available from the hydro- It is suggested that the lake drained via an R-channel which power gauging and water catchment station, located ∼1 km down- could have started opening a week before the event (Huss and stream the glacier outlet by the time of the study (2005). The others, 2007). However, it is estimated that before the outburst discharge time series shows strong daily fluctuations and seasonal flood (13 June), the lake contribution prior to the main phase variations (Fig. 4). Note that proglacial discharge is in phase with of the drainage was stored sub- or englacially and is therefore the subglacial water pressure and thus we take this as the recharge not accounted for in our model. rate, in which our model directly enters the subglacial system. From the dataset, we select five snapshots across the melt sea- However, the recharge needs to be spatially partitioned between son. We chose the timing of each snapshot such that it is at the the moulins on the glacier trunk (which is the model domain) end of a relatively stationary period, in order to make it represen- and between the tributaries (input via inflow boundary condi- tative of a stable situation of the subglacial system (Fig. 4 and tions). We do this partitioning according to the melt rates calcu- Supplementary Table S1). lated by a calibrated, distributed glacier surface melt model (Huss and others, 2008). According to the results, the trunk of Gorner 3.2 Modelling choices and settings Glacier (i.e. the area investigated in this study) contributes ∼22% of the total discharge during the summer months, whereas Total water recharge in the subglacial system is set to the dis- the remaining catchment runoff stems from higher regions of the charge measured at the outlet for each snapshot (Fig. 4). The rela- catchment and the tributary glaciers. tive contribution of each tributary is extracted from the Dye-tracer tests were carried out throughout the melt season distributed surface melt model results and input from tributary from four different moulins (m1–m4, Fig. 3). Tracer-transit glaciers as Neumann boundary conditions. In addition, basal times were extracted from the breakthrough curves of dye-tracer melt, which is negligible compared to the total recharge, is − tests and corrected for proglacial residence time following assumed to correspond to a uniform 5 mm a 1, following the Werder and others (2009). Overall, tracer-transit times for most same order of magnitude as presented in Herman and others of the injection points across the season ranged between 2 and (2011) and Cuffey and Paterson (2010). Water exits the subglacial 4 hour. The only exception corresponds to moulin m2, the most system at the glacier outlet, which is modelled as a Dirichlet distant to the glacier outlet, which showed a transit time of boundary condition at atmospheric pressure. ∼13 hour in June and September 2005. Tracer-transit times are As precise moulin locations and water recharge are unknown, presented in Figure 4 as green vertical bands, defined by the injec- sensitivity tests are performed to evaluate their influence in the tion time and the time of exit at the glacier outlet. Water flux subglacial channel generator. Here, we carry out a sensitivity entering the moulins was not measured during the tracer experi- test evaluating the unconditioned networks as it allows inspecting ments. In addition, a mapping of moulins location was carried out the channel networks before running the more computationally in the field in 2005. We expect that several moulins may have expensive water flow model. been missed by the survey, but it is likely that their spatial density First, the subglacial channel generator is used to produce mul- is well represented. tiple realizations of unconditioned subglacial systems by ran- In addition, the seasonal drainage of a large ice-marginal lake domly sampling structural parameters (lxy,s,f). In addition, we impacts on the evolution of the subglacial system (Fig. 3). Note explore the influence of the variance of the GRF on the network that lake drainage was observed between 13 and 15 June 2005. structure (Eqn (2)) and set it similar to small-scale changes in the

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hydraulic potential because the variance should not modify the out in Alpine glaciers have shown that transit speed (straight general trend given by the other terms in Eqn (2). This step allows path from the injection point to the glacier’s snout divided by − defining the structural parameter ranges where networks are transit time) rarely exceeds 1 m s 1 (e.g., Nienow and others, realistic. 1996; Schuler and others, 2004; Werder and others, 2010; Second, we construct a moulin probability density map from a Chandler and others, 2013). Indeed, tracer tests carried out in survey carried out in 2005 (Loye, 2006). Third, we randomly sam- 2005 from four different moulins (Fig. 3) resulted in transit − ple moulin locations from the density map (Supplementary speeds in the range from 0.1 to 1 ms 1 (Werder and others, Fig. S1) and assign a water recharge contribution proportionally 2010). Thus, we chose to bound tracer-transit speeds at the mou- to the Voronoi cell area of each moulin. Several random realiza- lins with observations to a range of dts and dts (Eqn (11)) cor- min −max tions including different number of moulins were used to deter- responding to transit speeds of 0.1 and 1.2 m s 1. Note that transit mine the consequences of such assumptions (Supplementary speed is computed from the total residence time, including inflow Figs S2–S5). To compare the different channel network realiza- modulation time. Water recharge into the moulin as well as mou- tions (e.g., Fig. 5a–d), we compute the likelihood for each cell lin volume is unknown, hence inflow modulation time is to be occupied by a channel weighted by the accumulated water unknown and thus we incorporate a fitting parameter that τ flow (e.g., Fig. 5e, f), obtained by routing the hydraulic potential accounts for inflow modulation time ( i), with a range from 0 (Eqn (2)). Note that the normalized accumulated flow differs to 20 h. This is critical in order to avoid model overfitting. For from the number of upstream-cells as is commonly found in transit speeds that fall outside the physically reasonable range, ts −1 the literature (e.g., Arnold and others, 1998). Here, the accumu- we consider a standard deviation of si = 0.25 m s (Eqn (11)). lated flow is weighted by the pointwise recharge from the moulins Water pressure exceeding surface elevation has only rarely and tributary glaciers. Overall, sensitivity tests found that moulin been observed on glaciers and has not been reported for Gorner number, location and relative recharge does not greatly impact the Glacier. Here we decrease the likelihood of a model when the channel network structure (see Supplementary material). This can hydraulic potential exceeds the surface elevation. This is done be explained by the fact that in this case study, water recharge into by evaluating Eqn (12) at 108 points dispersed at regular 250 m moulins within the model domain only accounts for 22% of the intervals over the entire surface of the glacier and using a standard z z total subglacial discharge. Moreover, unconditioned channel loca- deviation of si = 10 m. The value of si has been chosen to rap- tions for all configurations are well constrained by ϕ* and result in idly decrease the likelihood of water pressures exceeding this one or two main channels receiving most of the accumulated flow. bound. However, we note that this term plays a secondary role For the following analysis and inversion, we thus run the model by driving the model towards a solution closer to the actual bore- using a single set of 60 moulins whose positions are randomly hole observations. sampled from the moulin probability map. A selection of uncon- Lastly, we select snapshot dates across the season as well as a ditioned channel network realizations is presented in Figure 5. reference snapshot. The snapshots are defined as the average con- Next, the parameter inversion strategy requires defining the dition at midday within a 15 minutes window. The main criteria misfit function of each component (Eqns (10–12)). Here, two to select snapshot dates is to find periods where water pressure, sources of errors related to borehole observations are considered: discharge and meteorological variables are as stationary as pos- borehole location (or spatial representativeness of a punctual sible. Then the snapshot is considered at the end of the stationary measurement in a grid) and water pressure. As model errors are period. It is likely that the channel network is well developed in unknown and are usually larger than the instrumental error, we midsummer. We thus select snapshot 3 as the reference snapshot. considered borehole water level standard deviations However, tests (not shown) indicate that the choice of the refer- pw = si 10 m, and spatial tolerance of rbh = 100 m (Eqn (10)). ence snapshot does not greatly influence the results. As shown Tracer-transit times are largely dependent on water recharge in Figure 2, each snapshot is an independent inversion where into moulins and their geometry (e.g., Werder and others, the structural parameters are bounded in the vicinity of the 2010). As these data are not available, tracer-transit times are selected CS system structural parameters (lxy,f,s). The bounds only considered as a measure of physically reasonable velocities considered are s ±50m, lxy ± 50 m and f ± 0.1. A summary of in the subglacial system. In general, tracer experiments carried the data used for the inversion is presented in Table 1.

Fig. 5. (a–d) Examples of unconditioned channel networks. (e) Colour scale shows the normalized sum of the accumulated flow for 500 realizations. (f) Colour scale shows the logarithm base 10 of the normalized sum of accumulated flow for 500 realizations.

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Table 1. Model coefficients and parameters boreholes showing diurnal oscillations in water pressure (bh1, bh3 and bh4). Boreholes with a high and steady hydraulic head are Parameter Definition Value/bounds Units located in areas distant from predominant channels. One of the Model constants most significant differences between the inversions is the point −3 ρw Water density 1000 kg m −3 where the main channels merge: for example, CS 1 has two ρi Ice density 917 kg m − g Gravitational acceleration 9.81 m s 2 main parallel channels that join near the glacier outlet whereas B bedrock elevation M CS 2 has only one predominant channel already 3.5 km in front H Ice thickness m of the glacier snout (a). Note that for CS 3 parameters a and b Derived from geometry and constants converged to a range that differs from the rest of the models. ϕ z Elevation potential MPa The a and b parameter values indicate that CS 3 presents smaller pi Ice overburden pressure MPa f∗ Hydraulic potential from Shreve’s MPa radii for channel starting points and larger radii near the outlet in equation comparison to other CS. However, note that the sum of the vol- Water flow model variables ume of all subglacial channels (TCV) in CS 3 is similar to CS 6, ϕ Hydraulic potential MPa indicating that the channel networks of CS 3 and CS 6 share simi- pw Water pressure MPa N Effective pressure MPa lar overall volumes. This difference indicates that the channel Variables determined by inversion procedure structures influence the hydraulic efficiency of the network and r Channel radius 0–20 m therefore particular channels radii are needed for conditioning a −10– −2.5 2 −1 Td Transmissivity distributed system 10 10 m s the model to observations. τ a – i Inflow modulation time 0 20 h Borehole water pressure residuals show that most observations ϕR Gaussian random field perturbation MPa Channel generator variables lie within the uncertainty bounds (Fig. 6b), except for bh3, which a Radius scaling factor 0.01–5mis difficult to match because it implies a sharp hydraulic gradient. b Radius hierarchical order factor 0.01–4 Indeed, boreholes 2 and 3 are at a distance of 220 m, but show a – lxy Integral scale 150 750 m dissimilar water pressure level and behaviour. The hydraulic para- f Flotation factor 0.8–1.1 s Shift parameter 0–32.5 km meters (a, b and Td) of the CS systems are well constrained. Parameter inference However, these parameters differ from one conditioned model j S Misfit function for each data type to another. The latter implies that each channel network architec- [ , , b where j pw tt z ture has a particular geostatistical radii relation (Eqn (4)) that best p si w Standard deviation of the error in 10 m fits the observations. The transmissivity (Td) of the maximum borehole water pressure − − − likelihood models ranges between 10 7.9 and 10 7.2 m2 s 1. rbh Distance tolerance for water 100 m pressure borehole location Lastly, the structural parameters (lxy,s,f) converged to a different z si Standard deviation of the error in 10 m value for each CS, showing that each network is different by con- hydraulic head exceeding glacier struction (c). surface elevation. zi Surface elevation m ts −1 si Standard deviations of the errors in 0.25 m s transit speed ts −1 4.2 Inversion of subglacial systems across the melt season dmin − max Bound for speed in moulins 0.1–1.2 m s

aCorresponds to a variable of the subglacial channel generator as well. We selected CS 1, as the maximum likelihood CS system model to bSuperscripts correspond to pw water pressure, ts transit speed and z hydraulic head carry on the inversion across the melt season (snapshots 1–5). exceeding glacier surface elevation. This choice is supported by its high likelihood and the fact that two other independent runs (CS 3 and Supplementary Fig. S8) converged to a similar network structure. Additionally, we per- 4. Results formed the inversion across the melt season assuming channel network persistence of CS 2 (Supplementary Fig. S9). 4.1 Reference snapshot inversion (snapshot 3) Inversion results show that the maximum likelihood CS system A set of 15 independent inversions were run for the reference of the five snapshots presented, as expected, minimum variations snapshot (snapshot 3). Each inversion is configured to start in the network architecture throughout the melt season (Fig. 7a, c). with a different set of parameters. From the 15 independent inver- Similarly, the CS systems present channels in areas with lower sions, 11 different local maxima or CS systems were found (some hydraulic potential, separated by high-pressure areas of the dis- independent inversions converged to the same local maxima). tributed system. Note that for each of the inversions a likelihood distribution of Across all snapshots, a majority of borehole pressure observa- the inferred parameters is calculated by our numerical scheme. tions are within the uncertainty bounds of the model (Fig. 7b). The maximum likelihood CS systems are shown in Figure 6a Note that snapshots 2, 3 and 4 present a similar magnitude in (see Supplementary Fig. S7 for additional results). Note that all water recharge, however, the observed borehole water pressure but one inversion had a similar maximum likelihood value, differs for each snapshot (Fig. 7b). The channel radius parameters whereas the remaining inversion converged to significantly (a, b) are subject to relatively small differences, which allow hon- lower likelihood value (CS 9 in Fig. S7). Overall, it was found ouring the observations of each snapshot (Fig. 7c). Snapshots 4 that water flow occurs predominantly in channels that run-in and 5 indicate similar pressure observations, however, the depressions of the hydraulic potential (e.g., CS 1–5) and areas recharge forcing of snapshot 4 is more than triple that of snapshot between channels are subject to a relatively higher hydraulic 5. Consequently, parameter a (radii linear scaling) is smaller in potential. CS system CS 6 does not fully capture this pattern, as snapshot 5. This is clearly indicated by the total volume of subgla- its southern channel is not located in a depression of the cial channel networks (TCV), where the TCV of snapshots 2, 3 hydraulic potential (Fig. 6a). Note that CS 6 transmissivity is rela- and 4 is more than double of that for snapshot 5. The transmis- tively large compared to other CS, implying that more flow occurs sivity of the distributed system does not vary significantly across in the distributed system and consequently the hydraulic potential the inferred snapshots. As the structural parameters have been field is smoother in comparison to other CS (Fig. 6c). All CS sys- bounded, lxy,sand f do not show any major difference between tems exhibit the presence of high discharge channels passing near the snapshots.

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Fig. 6. Conditioned subglacial systems (CS) for six independent inversions of the reference snapshot (snapshot 3). CS 1 is highlighted in grey as it is propagated throughout the rest of the melt season. (a) Maximum likelihood of CS hydraulic potential field and channel discharge. Note that CS 1, 3 and 6 present two main parallel channels, whereas CS 2 and 5 present one dominant channel downstream of the boreholes. (b) Borehole water pressure residuals (boreholes in x-axis). Observations are displayed as blue vertical bands and black dot indicates the maximum likelihood model displayed in (a). Boxplots indicate the uncertainty with boxes corresponding to the 25th and 75th percentiles, respectively. The whiskers extend to the most extreme data points (∼±2.7σ for normal distribution). Outliers are plotted as red cross. (c) Inferred parameter distributions (CS in x-axis). Black dots indicate the maximum likelihood model displayed in (a), boxplots the par- ameter residuals. The total volume of subglacial channels (TCV) is shown in the right-hand side in m3.

5. Discussion on building a parsimonious model that allows the exploration of In this study, we infer properties of the subglacial drainage system multiple solutions to explain observations. In this section we of Gorner Glacier based on fitting our model to observations at explore the limitations, advantages and further work associated different snapshots across the melt season. Our approach focuses with our framework.

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Fig. 7. Conditioned subglacial system (CS 1) across the melt season (five snapshots). Grey bands indicate reference snapshot (snapshot 3). (a) Maximum likelihood hydraulic potential field and channel discharge. Discharge at the outlet is provided in parenthesis (b) Borehole water pressure residuals (boreholes in x-axis). In snapshot 1, boreholes 4 and 5 do not have data. Observations are displayed as blue vertical bands and black dot indicates the maximum likelihood model dis- played in (a). Boxplots indicate the uncertainty with boxes corresponding to the 25th and 75th percentiles, respectively. The whiskers extend to the most extreme data points (∼±2.7σ for normal distribution). Outliers are plotted as red cross. (c) Parameter distributions (CS in x-axis). Black dots indicate the maximum likelihood model displayed in (a), boxplots the parameter residuals. The total volume of subglacial channels (TCV) is shown in the right-hand side in m3.

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The proposed model makes the following necessary simplifica- We designed a two-step inversion strategy that first allows infer- tions: (1) it assumes a spatially uniform transmissivity to represent ring multiple subglacial systems models of a reference snapshot. the distributed system, (2) it produces the channel network structure Here we selected snapshot 3 (10 August 2005), i.e. a midsummer by a flow routing method, (3) it uses a statistical relation for the date, as it is more likely that channel networks are well developed. channel radii, (4) it assumes constant recharge for the duration of For this date, we obtained different CS systems which honour obser- a snapshot period, which is necessary to resolve water flow in steady vations, demonstrating that data are insufficient to constrain a well- state. Conservatively, we selected the snapshots after a relatively defined or unique channel network. Secondly, we selected two CS stable recharge period in order to be relatively close to steady-state systems (CS 1 maximum likelihood model, and CS 2 in conditions. These assumptions make the model computationally Supplementary material) to force the results of following snapshots inexpensive, and as a result, its parameters can be inferred from across the melt season to resemble their conditioned system. The data, along with uncertainty quantification. assumption that the channel network’s main structure persists An important challenge when applying our model regards the across the melt season is practical, as it avoids inversion parameters uncertainties in recharge conditions. We use a distributed melt that introduce discontinuities in the likelihood function. As a con- model calibrated with measured discharge and in situ ablation mea- sequence, convergence is much more easily reached in the subse- surements, to estimate how recharge is partitioned between different quent snapshots. Moreover, the hydraulic parameter variability locations of the model domain. Tributaries account for 76% of the found within the reference snapshot is large (see Fig. 6)andthus total recharge and moulins account for 22% of the recharge. The considering one channel network allows for a better understanding remaining 2% of the discharge is incorporated outside the model of changes on the hydraulic parameters (enlargement and shirking domain (between the glacier snout and the gauging station). To of channel radii) across the melt season. However, this strategy cer- assess the impact of the subglacial channel generator on the number tainly underestimates uncertainty as it only explores few models of moulins, location and relative contribution, we carried out a sen- across the melt season. Further work, with more computational sitivity analysis. It was found that these factors do not greatly influ- resources, should implement inversion frameworks to infer inde- ence the network structure (see Supplementary material). This pendently each snapshot (or all snapshot simultaneously), as well result, however, cannot be generalized, as it is particular to this as not enforcing channel network persistence across the melt season. site where channels are well constrained by Shreve’shydraulic In that case, channel network persistence across the melt season will potential and moulin recharge corresponds to only about a fifth rely on data and its uncertainties of the total recharge. One particularity of Gorner Glacier is its poly- Subglacial transit times from four moulins were used to con- thermal nature in the ablation zone (Ryser and others, 2013). The strain the channel network parameters to a physically reasonable cold ice body acts as an impermeable barrier for meltwater except range. A number of assumptions, which were made to avoid when water flows continuously into pre-existing moulins or cracks. model overfitting, needed to be made as the available tracer obser- Such moulins in the cold ice area are rare, hence, they drain a larger vations only loosely constrain the transit times: using a large range surface area providing sufficient meltwater to maintain them. This for the inflow modulation time (0–20 hour), as well as providing feature could imply a more stable supra- and englacial system, extended bounds for the subglacial transit speeds through channels − and consequently that recharge locations into the subglacial systems (0.1–1.2 m s 1). Nonetheless, including even these broad con- are persistent across the melt season. straints on subglacial transit and inflow modulation facilitated Results and previous studies have stressed that flow routing model space exploration and convergence. Experiments performed models are sensitive to small-scale perturbations (e.g., Chu and without these constraints (not shown) converged to subglacial others, 2016; Irarrazaval and others, 2019). This is particularly models composed of small channels dominated by large transmis- relevant when downscaling (or interpolating) bedrock elevation sivities in the distributed system. This configuration resulted in data to match the higher spatial resolution of a surface DEM, as unrealistically fast water transit speeds and a smooth hydraulic flow routing becomes biased towards small-scale features from potential field that did not capture sharp hydraulic potential gradi- the high-resolution DEM. Moreover, modelled bedrock elevation ents observed in nearby boreholes. Clearly, if a more accurate solu- can significantly differ depending on the model and available data tion (which we found) exists in the transit-time constrained space, used to determine the ice thickness (e.g., Farinotti and others, it also exists in the unconstrained space, but it appears that the cap- 2017). The subglacial system generator adds variability, which ability of the DREAM(zs) algorithm to find it is reduced. Thus, in allows generating sets of unconditioned subglacial systems. The our case, limiting transit speeds facilitated convergence to local core of the channel network generation is based on the addition maxima and forced discharge into channels. However, other pos- of a nonuniform spatial perturbation term, modelled as a GRF, sible strategies could have been considered such as adding a to the hydraulic potential field. The GRF is parametrized by struc- lower limit for channel radii, lowering the upper bound of Td or tural parameters lxy,,fand s, allowing for a low dimensionality of utilizing another algorithm to explore the model space. the inverse problem. As a result, modifying structural parameters The high sensitivity of the inversion to subglacial transit time lxy,fand s results in global changes in the network. A drawback of constraints also highlights that having more accurate observations this approach is that fitting independently individual water pres- available would constrain the inversion significantly better. This sure observations requires locally modifying a channel, which is finding is coherent with previous work of Irarrazaval and others not possible with only three global parameters. To avoid overfit- (2019) on a synthetic inversion, where tracer tests considerably ting, we add a spatial tolerance to the borehole locations (Eqn reduced the uncertainty of the model parameters. This means (8)), allowing for a broader range of networks to match observa- that future tracer experiments need to separate total transit time tions. Although the choice of adding a GRF to Shreve’s equation into inflow modulation time and subglacial transit time, which satisfied the objectives of this case study, further work should necessitates observations of moulin recharge and obtaining con- investigate the variability incorporated with different parametriza- straints of moulin geometry (e.g., Nienow and others, 1996; tions or directly using a nonspatially uniform flotation factor, for Schuler and others, 2004; Werder and others, 2010). example. Another drawback is that the exploration of the model Conditional subglacial systems show that water is concentrated space is challenged by discontinuities in the likelihood function in subglacial channels with low water pressure surrounded by as a consequence of modifying parameters (lxy, s, f ). This points areas of poor connectivity and high-water pressure. The subglacial out the importance of having a fast forward model which enables channels were enlarged at the beginning of the melt season (from a large number of runs necessary for uncertainty quantification. snapshot 1 to snapshot 2), maintaining their total volume over the

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summer (from snapshot 2 to snapshot 4) then substantially Alpiq SA. We thank Associate Chief Editor Ralf Greve, Scientific Editor Ian shrink at the end of the season (from snapshot 4 to snapshot Hewitt as well as reviewers Douglas Brinkerhoff and Martin Truffer for their con- 5) as shown in Figure 7c. This is compatible with current under- structive comments that led to improve this publication. standing of the seasonal evolution of the glacier drainage system (e.g., Fountain and Walder, 1998). References The result of our framework is a set of solutions for subglacial systems that can match the data with a high likelihood (Fig. 6). Arnold N, Richards K, Willis I and Sharp M (1998) Initial results from a dis- The set of inferred subglacial systems depends on the definition tributed, physically based model of glacier hydrology. 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Nature Geoscience 6(3), found that varying the spatially homogeneous flotation factor in – ’ 195 198. doi: 10.1038/ngeo1737. Shreve s equation can divert subglacial flow routing paths (e.g., Chu W, Creyts TT and Bell RE (2016) Rerouting of subglacial water flow Banwell and others, 2013; Chu and others, 2016). Here we between neighboring glaciers in west Greenland. Journal of Geophysical added a GRF that accounts for a spatially nonuniform term. Research: Earth Surface 121(5), 925–938. doi: 10.1002/2015JF003705. The choice of a GRF is practical in an inversion framework, as Church G and 5 others (2019) Detecting and characterising an englacial con- it reduces the dimensionality of an unknown 2-D spatial field. duit network within a temperate Swiss glacier using active seismic, ground Consequently, the incorporation of a GRF adds variability to penetrating radar and borehole analysis. 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Journal of Glaciology 64(248), 897–916. doi: 10. but data are still too sparse to result in one single solution. This 1017/jog.2018.78. highlights the importance of working in a multi-solution frame- Farinotti D and 5 others (2017) How accurate are estimates of glacier ice work, particularly when using results for prediction as distinct thickness? Results from ITMIX, the ice thickness models intercomparison channel networks could influence, for instance, basal sliding or eXperiment. The Cryosphere 11(2), 949–970. doi: 10.5194/tc-11-949-2017. respond differently under a glacial lake outburst flood scenario. Farinotti D, Huss M, Bauder A and Funk M (2009) An estimate of the glacier – Benefit of the use of tracer experiment data for uncertainty ice volume in the Swiss Alps. Global and Planetary Change 68(3), 225 231. reduction was limited. To avoid overfitting, tracer experiments doi: 10.1016/j.gloplacha.2009.05.004. 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(and more parameters) versus physically simpler models capable doi: 10.1002/2016GL068337. of multi-solution, uncertainty quantification and data integration. Gulley JD and 5 others (2014) Large values of hydraulic roughness in subgla- cial conduits during conduit enlargement: implications for modeling con- – Supplementary material. The supplementary material for this article can duit evolution. Earth Surface Processes and Landforms 39(3), 296 310. be found at https://doi.org/10.1017/jog.2020.116. doi: 10.1002/esp.3447. Herman F, Beaud F, Champagnac J-D, Lemieux J-M and Sternai P (2011) Acknowledgements. I.I. was funded by the National Agency for Research and Glacial hydrology and erosion patterns: a mechanism for carving glacial val- Development (ANID), BECAS CHILE/2015–72160092. Proglacial discharge data leys. Earth and Planetary Science Letters 310(3–4), 498–508. doi: 10.1016/j. were provided by Grande Dixence SA, with the support of Hydroexploitation and epsl.2011.08.022.

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